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Citation for the original published paper (version of record):
Terschlüsen, J., Agåker, M., Svanqvist, M., Plogmaker, S., Nordgren, J. et al. (2014)
Measuring the temporal coherence of a high harmonic generation setup employing a Fourier
transform spectrometer for the VUV/XUV.
Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers,
Detectors and Associated Equipment, 768: 84-88
http://dx.doi.org/10.1016/j.nima.2014.09.040
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1
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Measuring the temporal coherence of a high harmonic generation
setup employing a Fourier transform spectrometer for the VUV/XUV
3
4
J.A. Terschlüsen*1, M. Agåker1, M. Svanqvist1,2, S. Plogmaker1, J. Nordgren1, J.-E. Rubensson1,
5
H. Siegbahn1, J. Söderström1
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1
7
Department of Physics and Astronomy, Uppsala University, Box 516, 75120 Uppsala,
Sweden
8
9
2
Present address: Defense & Security, Systems and Technology Department, FOI Swedish
Defense Research Agency, 14725 Tumba, Sweden
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Abstract
14
In this experiment we used an 800 nm laser to generate high-order harmonics in a gas cell
15
filled with Argon. Of those photons, a harmonic with 42 eV was selected by using a time-
16
preserving grating monochromator. Employing a modified Mach-Zehnder type Fourier
17
transform spectrometer for the VUV/XUV it was possible to measure the temporal
18
coherence of the selected photons to about 6 fs. We demonstrated that not only could this
19
kind of measurement be performed with a Fourier transform spectrometer, but also with
20
some spatial resolution without modifying the XUV source or the spectrometer.
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22
23
Keywords
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25
26
27
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Fourier transform spectrometer
HHG
Temporal coherence
Spatial coherence
Soft X-ray
XUV
*Corresponding author.
E-mail address: [email protected] (Joachim Terschlüsen)
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1. Introduction
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The measurement of temporal coherence of visible or infrared radiation can be done
35
employing equipment like a Michelson interferometer [1]. In a Michelson interferometer the
36
beam is split in two components, where one of them is delayed with respect to the other by
37
changing the path length in one of the interferometer arms. Both beams are then
38
recombined and the resulting interference pattern is recorded as a function of path length
39
difference. By scanning the delay and observing the changes of the interference fringes, one
40
can measure the field autocorrelation of the radiation [2]. From that one can get information
41
about the temporal coherence as well as the spectrum of the radiation. Temporal coherence
42
is seen in the presence or absence of interference between the two overlapping beams
43
while spectral information can be retrieved through a Fourier transformation of the
44
recorded interference fringes as function of path length difference [2]. Although the
45
principle is the same for shorter wavelengths in the Vacuum Ultraviolet (VUV) or Extreme
46
Ultraviolet (XUV) regime, the implementation is much more difficult. Key reasons for this are
47
that VUV/XUV radiation is strongly absorbed by all kinds of materials, even air, making it
48
necessary to work in vacuum and to use only reflective optics as well as special radiation
49
detectors. Furthermore, one has to keep the angle of incidence large enough to achieve total
50
external reflection, since reflectivity at normal incidence is very poor at these photon
51
energies [3]. This also implies that the number of reflections should be kept at a minimum.
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Additionally, the requirements on the surface quality of the optics increase with decreasing
53
wavelength, and so do the demands on mechanical stability and positioning precision of
54
movable parts.
55
Although the demands on a setup rise with the photon energy, the temporal coherence of
56
XUV pulses has been measured. This was done e.g. for Free Electron Lasers (FELs) pulses
57
with schemes employing wave front dividing beam splitters [4,5] or for collisionally pumped
58
soft-x-ray lasers using multilayer intensity dividing beam splitters [6].
59
To measure the temporal coherence of XUV radiation produced in a High Harmonic
60
Generation (HHG) process [7], one can use dedicated setups as done in Refs. [8,9]. In these
61
setups the driving laser beam is split prior to the HHG generation in a semitransparent
62
intensity-dividing beam splitter. These two phase-locked beams are then used to generate
63
two separate XUV sources in the same gas very close to each other. Hence, the XUV
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radiation from both sources is also phase-locked and the coherence properties can be
65
analyzed by overlapping the two XUV beams. However, this method cannot be used easily at
-2-
66
existing sources and beamlines due to limitations in laser power and accessibility of the
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setups.
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The development of wave front dividing beam splitting methods made a recent
69
development of Fourier transform spectroscopy into the VUV and XUV region possible
70
[10,11]. This allows Fourier transform spectrometers to be used in the VUV and XUV range
71
to analyze the so-called visibility of the interference pattern in the same way as can be done
72
for visible and infrared sources. Thus, the measurement of the temporal coherence of a
73
source, without the need of modifying it is possible.
74
A Fourier transform spectrometer, although capable of measuring the coherence length, is
75
however not capable of measuring the pulse length. A proper measurement of the pulse
76
length is a topic of its own and is discussed for optical wavelengths in detail e.g. in Ref. [12].
77
The major difference is that one needs to measure not the field autocorrelation of the signal
78
but the intensity autocorrelation which requires a second order process. Such a second
79
order process requires high light intensity and adds additional complexity which makes the
80
measurement of the pulse length unachievable with a conventional Fourier transform
81
spectrometer.
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83
84
2. EXPERIMENTAL
85
In order to demonstrate the capability of using a Fourier transform spectrometer (FTS) to
86
measure the temporal coherence of short XUV pulses, the HELIOS (High Energy Laser
87
Induced Overtone Source) HHG light source at Uppsala University [S. Plogmaker et al. (in
88
manuscript)], and an in-house developed Mach-Zehnder type interferometer for the XUV
89
radiation [11] were used. Briefly, HELIOS is an HHG setup which uses an amplified
90
commercial Ti:sapphire based laser system (Coherent Inc.) with a center wavelength of 800
91
nm and a pulse duration of 35 fs. The repetition rate of the system is 5 kHz resulting in a
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total output power of 12.5 W. The laser is focused into a gas cell filled with Argon or Neon to
93
generate XUV radiation. The generated radiation is inherently coherent [13] and the upper
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limit of its pulse duration is set by the pulse duration of the driving laser. The energies that
95
are generated at HELIOS today are in the order of 20 eV to 70 eV and, since the radiation is
96
generated in noble gases, the harmonics are limited to the odd harmonics of the energy of
97
the driving laser photons.
-3-
98
Since several harmonics are generated in the HHG process at the same time a
99
monochromator is used to select a single harmonic. A traditional monochromator design
100
would prolong the pulse due to the path length difference introduced by every single grove
101
of the grating. To keep this prolongation small, HELIOS uses a monochromator design
102
employing a grating in a so called off-plane mount [14,15]. The HELIOS light source will be
103
discussed in further detail in a forthcoming publication [S. Plogmaker et al. (in manuscript)].
104
The spectrometer used in this experiment is a FTS of modified Mach-Zehnder type [11]. It is
105
equipped with a large-aperture comb-like wave-front-dividing beam splitter, made from a
106
super polished single crystal silicon mirror. An identically slotted mirror is used as a beam
107
mixer. The FTS was earlier tested at the I3 beamline [16] at the Max IV laboratory, Lund,
108
Sweden, both in direct [17] and indirect detection of the beam, and has been shown to work
109
at least up to 55 eV photon energy.
FIG. 1:
Schematic drawing of the HELIOS source with the Fourier transform spectrometer
attached behind the monochromator.
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In the current experiment, Argon was used in the gas cell and the FTS was mounted directly
111
behind the monochromator allowing measurements of the temporal coherence of a single
112
harmonic of the HHG radiation. A schematic drawing of the setup used in the experiment is
113
shown in FIG. 1. A typical image of the XUV radiation on the detector of the FTS can be seen
FIG. 2:
FTS detector image of the 27th harmonic. The path length difference is 1.35 µm
(corresponding to 45.1 fs) for a) while it is zero µm for b). Hence, interference
fringes are visible in b) but not in a). See text for details about the interference
pattern.
114
in FIG. 2. Note that the path length difference in FIG. 2 a) is larger than the temporal
115
coherence and hence no interference pattern can be seen. The fringes visible in FIG. 2 a) are
116
static and do not change when the path length difference between the arms of the
117
spectrometer is varied. This intensity pattern has its origin rather in the light passing first the
118
beam splitter and then the beam mixer which both act as multi slits. The fringe pattern
119
caused by the interference at zero path length difference can be seen in FIG. 2 b).
120
Additionally, in both FIG. 2 a) and b), one can see a grid like structure which probably
121
originates from a meshed aluminum foil in the monochromator that is used to block the
122
infrared driving laser while transmitting the XUV radiation.
123
124
3. Results
-5-
125
The temporal coherence can be determined by measuring the visibility of the interference
126
fringes when changing the delay between the two beam paths in the spectrometer. The
127
visibility Vb is defined as [18]:
𝑉𝑏 =
𝐼𝑚𝑎𝑥 − 𝐼𝑚𝑖𝑛
𝐼𝑚𝑎𝑥 + 𝐼𝑚𝑖𝑛
(1)
128
where Imax and Imin are the intensities of constructive and destructive interference of the
129
observed fringes. The temporal coherence 𝜏𝑐 of the radiation can be calculated from the
130
visibility as [18]
∞
𝜏𝑐 = ∫ |𝛾(𝜏)|2 𝑑𝜏
(2)
−∞
131
where 𝛾(𝜏) is the complex degree of coherence which can be expressed as
𝑉(𝜏)
𝛾(𝜏) =
𝑉(0)
132
and 𝜏 is the traveling-time difference for the light in the two arms of the spectrometer. V
133
denotes the visibility Vb, but with a constant background subtracted. Note that the
134
subtraction of background is necessary due to noise in the signal. This noise leads to an
135
offset of the visibility Vb when it increases the intensity of an interference maximum Imax but
136
decreases the intensity of an interference minimum Imin leading to an overestimation of the
137
visibility Vb. When referring to the visibility from here on we refer to V and not to Vb (unless
138
explicitly stated differently).
139
The way the temporal coherence was defined in equ. (2) is the same as used for measuring
140
the longitudinal coherence of FLASH by Schlotter et al. [4]. This definition was introduced by
141
Manel 1959 and its importance was exemplified in [19].
142
This definition bases the temporal coherence on an integral over the squared complex
143
degree of coherence 𝛾(𝜏). That way, one gets rather independent of the shape of the
144
frequency spectrum, the temporal shape of the pulse or its chirp. This is an advantage
145
compared to using peak-shape dependent parameters as full width at half maximum
146
(FWHM), especially since one can see in FIG. 4 that the peak shape of the visibility seems to
147
be more complex than a Gaussian peak.
-6-
FIG. 3:
Intensity of a single pixel out of the interference pattern of the 27th harmonic
(~42 eV) in gray plotted against the pulse delay. A zoomed cutout of a) is
displayed in b), showing how the interference pattern consists of single
interference fringes. Black dots denote measured points, gray and white
modulations of the background show the slices used to gain Imax and Imin. The
blue line/bars in a)/b) show the upper envelope used as Imax while the red
line/bars show the lower envelope used as Imin.
148
To measure the temporal coherence of the XUV radiation behind the monochromator (see
149
FIG. 1) the intensities of the interference patterns were recorded as a function of path length
150
difference by recording the full 2D detector image while scanning the path length difference
151
in the spectrometer. In this way, the intensity progression of every pixel of the detector
152
image was obtained. Such an intensity progression for a single representative pixel is shown
-7-
153
in figure FIG. 3. The successive intensity values of a single pixel were divided in slices of ten
154
measurements as can be seen in FIG. 3 b). One slice contains slightly more than one period
155
of the interference fringes. The highest and lowest value within such a slice was assigned to
156
Imax and Imin respectively. That way, it was possible to calculate a visibility curve for Vb by
157
means of equ. (1) for every single pixel on the detector. Due to the slicing, the density of
158
successive values of the visibility was reduced by a factor of ten compared to the intensity
159
measurement. This results in one visibility value per 42 nm scanned path length difference
160
(corresponding to a delay of 0.14 fs). To obtain the visibility of a whole detector region, all
161
single-pixel visibilities in a specific region were averaged leading to a clear visibility
162
distribution although the interference pattern on the detector looks quite complicated.
163
Due to assigning the maximum and minimum value of ten measurements to Imax and Imin the
164
visibility Vb of a single pixel also contains the noise of the measurement. Since the
165
interference picture contains about 16000 pixels this noise adds up to a constant
166
background which can be subtracted. By subtracting this background one gets a background
167
free visibility V as discussed in the beginning of this chapter.
FIG. 4:
Measured visibility of the fringes of the 27th harmonic (42 eV) in red measured
on the whole detector image as shown in FIG. 2 plotted against the path
length difference of the spectrometer. A linear background has been
subtracted. The black line represents a fit consisting of one Gauss peak plotted
in light gray symmetrically surrounded by two smaller ones per side in dark
gray.
-8-
168
FIG. 4 shows this visibility V for the whole detector region plotted against the path length
169
difference in the spectrometer. The visibility curve can be fitted with good agreement by a
170
main Gaussian, which is symmetrically surrounded by two smaller Gauss peaks per side. The
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parameters for the fit can be found in
172
Table 1 together with the temporal coherence determined using equ. (2). The origin of this
173
structure is not known but could come from a pulse shape that is more complex than that of
174
a Gaussian.
175
Note that the visibility is unlikely to reach one, even for absolutely coherent radiation. This
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has its reason in the nature of the beam splitter and beam mixer in the FTS. Due to the multi
177
slit character of the beam splitter, the beams hitting the beam mixer will have a diffraction
178
pattern. This means that the intensity from each arm of the spectrometer does not
179
necessarily need to be equal for all parts of the detector. This is due to the fact that this ratio
180
depends on whether rather diffraction maxima or minima of the beam splitter are blocked
181
by non-reflective/non-transitive parts of the beam mixer.
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The spatially resolved detector image of the FTS makes it possible to measure the temporal
183
coherence not only for the whole beam at once but also spatially resolved. This was done for
184
seven subareas at different places of the detector as shown in FIG. 5. This analysis shows
185
that the visibility as well as temporal coherence differs at different areas of the beam. As
186
one can see, the visibility of the subareas is in all cases higher than the one on the full
187
detector. This behavior has its origin in the fact that the average intensity in the subareas is
188
higher than the average intensity on the whole detector image. This means that the dark
189
parts of the detector image have a rather low visibility of fringes, most due to the much
190
lower signal to noise ratio in them.
191
It is important to note that the numbers shown in
192
Table 1 should not be seen as quantitative values describing the exact behavior of the
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visibility in the single subareas of the detector regions. They are rather meant to show a
194
more qualitative picture of the visibility changes in FIG. 5. The fit parameters might vary
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quite drastically when repeating the measurement although the actual visibility curve does
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look quite similar. This behavior might have its origin in the large number of free parameters
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that are used to describe the visibility curve.
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The temporal coherences reported here are about half of the ones measured by Bellini et al.
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[8] for long electron trajectories in the HHG process. Furthermore, we did not detect a
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component of the radiation with much larger coherence length as expected in the middle of
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the HHG radiation cone originating from short electron trajectories in the HHG process [8].
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This might be due to uncertainties in the alignment of the spectrometer relative to the XUV
203
beam. On account of this, the amount of radiation originating from electrons on short
204
trajectories might have been reduced so much that their signal was drowned in the noise of
205
the detector.
- 10 -
FIG. 5:
a) Detector image of the 27th harmonic (42 eV). Green rectangles denote the
areas, which were analyzed separately to extract their visibilities. b) Visibilities
measured for the full detector image as well as the subareas from a) plotted in
dark red against the path length difference of the spectrometer. A linear
background was subtracted and each spectrum was vertically offsetted by
0.075 from the previous one for visual clarity. Light gray curves showing the
visibility of the full detector image are plotted to ease the comparison
between the subareas. The temporal coherences for each subarea are noted
to the left.
206
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207
Table 1:
Fit parameters obtained from fitting the visibility of the whole detector image of
208
the 27th harmonic (~42 eV) by a main Gaussian peak symmetrically surrounded
209
by two smaller ones. Note that the positions of the side peaks are at x0 + xs and
210
x0 - xs. The width σ denotes the standard deviation.
st
nd
temporal
main peak
1 side peak
2 side peak
coherence
amp. A pos. x0 width 𝜎
amp. A pos. xs width 𝜎
amp. A pos. xs width 𝜎
[fs]
[10−2 ]
5.5
1
areas
[fs]
[fs]
[10−2 ]
[fs]
[fs]
[10−2 ]
[fs]
[fs]
7.6
+0.000
2.7
0.85
8.5
1.6
0.65
14
8.0
6.6
9.0
-0.033
3.3
1.10
7.1
2.1
0.75
17
5.8
2
4.9
12.6
-0.070
2.6
1.70
8.3
1.6
0.75
18
8.2
3
5.2
12.8
-0.049
2.6
3.10
8.5
1.6
1.60
18
3.9
4
7.7
11.1
+0.054
3.6
2.50
11.0
3.0
0.82
15
5.7
5
5.4
12.1
+0.300
2.8
1.90
9.5
1.7
0.92
18
5.5
6
5.2
12.7
+0.310
2.7
2.00
8.9
2.1
0.71
21
6.1
7
6.4
11.3
+0.310
3.4
0.72
8.4
1.8
0.88
18
5.3
full
image
211
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4. Conclusion
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We have shown in the example of the 27th harmonic of HHG radiation (~42 eV) that a Fourier
214
transform spectrometer for the XUV can be used to directly gain knowledge about the
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temporal coherence of HHG pulses. In the future, one could ensure that the whole XUV
216
beam is imaged onto the middle of the detector of the Fourier transform spectrometer by
217
developing a new, more rigorous alignment process. This should allow distinguishing
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radiation with different coherence times at different parts in the XUV beam which was not
219
possible with the current beam alignment. Nevertheless, we have noticed a spatial variation
220
of the temporal coherence within the beam but on smaller scales than expected.
221
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Acknowledgements
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This work was supported by the Swedish Research Council (Vetenskapsrådet), the Knut and
224
Alice Wallenberg Foundation and the Carl Tryggers Foundation.
225
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